Plural grating spectrograph



Nov. 9, 1948. P. F. swlNGs PLURAL GRATING SPECTROGRAPH :5 sheets-sheet 1 Filed May 19. 1945 INVENTQR. Paz yaoleE/-T5n//NG5 Afro/amsn Nov' 91 P F. 'SWINGS l. l

,-PLURALGRATING SPECTROGRAPH Filed May 19, 1945 3 Sheets-Sheet 2 Afrox/vsn Nov. 9, 1948. I P. F. swlNGs 2,453,164

PLURAL GRATING SPECTROGRAPH v Fiul'ed May 19, 1945 s sheets-sheet s IN V EN TOR.

,00L raars A' .5w/N65 Afro/met and ample resolution are combined Patented Nov. 9, 1948 2,453,164 PLUaALl GRATING sPEcrRoGRAPH Polydore F.k Swings, Pasadena, Calif., assigner to Lane-Wells Company, Los Angeles, Calif., a corporation of Delaware Application May 19, 1945, Serial No. 594,656

This invention relates to spectroscopes and the like apparatus and, particularly, to spectrographs having wide spectral ranges, together with ample dispersion and resolution over the entire range.

' Industrial spectrographsfof both the high dispersion and low dispersion types aredecient in a number of aspects. The conventional high dispersion types of instrument` now available are able' to cover only a relatively small range of wave lengths, except at the penalty of extreme bulkiness.' Such conventional high dispersion instrumentsusually cover only the region from approximately 2,400 to 3,600 A, on one plate, and if other spectral regions are desired, adjustments and changes in the arrangement of the instrument elements must be made. For example, in a prism instrument. of the conventional type, the vglass prisms must be replaced by quartz prisms and the position of the lenses and the plate holder must be changed when shifting from the visual range of wave lengths to the ultra-violet. In a 13 Claims. (Cl. 88-14) a spectrograph of the character described, ca-" conventional grating instrument, either the angular position of the grating or the position of the plate holder, or that of the slit must be changed for the different spectral regions, and a number of successive exposures with accompanying readjustments have had to be made under these conditions if a. wide spectral range was to be covered. The taking of such successive exposures of the spectra can be only tolerated where perfect stability and constancy with respect to time of the source of illumination and material under investigation obtains during the interval of delay. Such perfection is seldom attained and Where the composition orcharacter of the material undergoing analysis is constantly changing 'I with time, asis often the case in certain processes, such successive `exposures separated by intervals of time required for modification of the spectrograph are not permissible.

Similarly, the conventional, low dispersion types of spectroscopes or spectrographs, such as .now commonly available, at best cover only` a relatively small part o f the spectral range useful in the usual application for which industrial instrulments may be employed.

It is, accordingly, an object of this invention to overcome the hereinbefore outlined disadvantages and limitations, inherent in the present conventional, industrial spectrographs, and to present an improvement in spectrographs, in which compactness, wide spectral range, high dispersion,

in one instrument.

n isanother object of lthis invention to provide A: r

`ure 1. f

pable of photographing a spectrum of wide spectral range in a single exposure. It is a still further object of this inventiont'o" provide a spectrograph of simplified and improved mechanical construction and operation. n

The objects of this invention are accomplished; in general, by'employing an arrangementof'two or more dispersion means, each giving: different and preferablylinearr dispersions o'f predetermined relative values, together with a single slit source of illumination and one or morefplate holders as desired, said dispersion means .being constructed and arranged so that two or more complementary or overlapping spectra having predetermined ranges and vrelative dispersion ratios may be simultaneously photographed at a single exposure if desired, `on asingle lmfor plate surface. More specifically, the objects of this invention are attained in apparatus employing the discovery that if two concave diffraction gratings having the same radius of curvaturebut vone hav.- ing twice as many ruled lines as the other one per unit length, are illuminated from a single slitsource of illumination, the resulting focused spectra can be conveniently made tocover both the ultra-violet and visible regions with the dispersion in the visible region being one half vof that in the ultra-violet and the focal planes-of the two spectra will lie on identical curves and in a common surface such that the two spectra can be photographed on the same rllm or plate, if desired. A plurality of gratings having numbers of lines bearing predetermined ratiosone to another may also be similarly employed to divide the spectrum into a plurality of complementary spectra as described hereinafter.

These and other objects, advantages, andi features of novelty will be evident hereinafter.

In the drawings, which show, by Way of illus;- 7tration, preferred embodiments of the invention Aand in which like `numerals, designate the, same or, similar parts throughout theseverall views;

Figure'l is a plan-sectional view of the spec- .trograph assembly, taken on line ,I--IiofFig- Figure 2 is a cross-.sectional elevation? of. the spectrograph assembly, taken on line 2 2 of Fig,-

Figure 3 is a schematic diagram 'showing the arrangementV in plan view of the elementsof the spectrograph of Figure l.

Figure4 is a perspective view ofthe elements ofv Figure 3. I"

. Figure 5 is asch'ematic diagram inl plan *View of an alternative arrangement of the elements of the spectrograph.

Figure 6 is a perspective view of the elements of Figure 5.

Referring now primarily to Figures 1-4, inclusive, IIJ is a rigid, hollow, base member approximately triangular in general plan form and having adjustable supporting and levelling screws I I.

Extending from the -forward end portion of' I9 which are, in turn, slidably supportedforrelative vertical adjustment at 2B and `2I on rod.

22,. extending upwardv from` the: basemember. I3; other conventional workholding" and illumination source means, may-be employed suited to thea particularY material to be. analyzed. As before.A stated; the. workholder assembly and the condenserv lens support resting on. their base members. I3 and I4 are. slidably supported on the' Ways I2.Y and are thusadjustable with respect to. one another, ,asf well.- as. with respect tothe slit holder 2,5, in order to obtain the, desired light intensity and, ifv desired, focus of the source; of illumination on the slit;

The light-analyzing elements` of. the spectrograph are housed: within. a. light-excluding'hous'- ing 'Ll-I supported on the upper surfaceof. the-base member' I0. The housing 2:1 comprises aforward endv member 23,` a rearward end member h. andan intermediate, interconnecting hood The-slitv holder 25and an integral beam splitter assemblyy 33. are supported, with their common axes parallel to the ways I2, inv an aperture 34 formed irr awall portion 35 ofthe forward end member 28 of the spectrograph housing. The

holdercomprises a generally cylindrical housing portion'` 3I` provided with aY forward opening 31.15 The opening 3i leads toa pai-r of oppositely positioned, vertically disposed knife-edge elements 4I and `42, which are adjustable-laterally in a pair'of inner, horizontally'extending recesses 38. and 39, to form a verticali, rectangular slitsource. of. illumination 43 of any desired width.

The:v beam splitter assembly portion 3'3" comprises a transparent', partial reector element 44 located on the commonaxis'45 ofthe slit holder 2.5 and. condensing Alens t6; and making ana-ngle of 45 therewithhis thereby' adapted to reflect a portion. of. the diverging, entering beam upward to the lower surface of an upper diagonal mir'- ror'4l,and thence. along. ari-upper path having a central beam axis 411' parallel'witlr axis 45. 'l'ltrev imrefiect'ed portionl of' the divergi'ng' light passing"through the.- partial' reector 44"' is reieetedafrom the. lowerrsurfaceof diagonal mirror 41 to the uppers/ urface` of'. a lower diagonal' mirror 49 and thence along a lower central beam axis" 4T which; is parallel with the upper beam axis 451. A pair of vertically: spaced', but'- identical slit-sources of illumination are thus in eii'ect ilormed.A from; a .single4 source, vfor .the two# dif fraction gratings hereinafter' described.

In the rear portion of the spectrograph housing and;- Within the end member 29" are a pair of concave diffraction gratings I and 52, supported vertical, alignment, one, above the other in 4 position to centrally intercept light owing from the beam splitter along the beforementioned upper and lower -parallel beam paths or aXes 41 and 41. The said concave gratings are adjustably supported upon a suitable structure comprising a generally rectangular frame 54 carrying upper and lower ways 55 and 56 upon which upper andl lower cells-51" and 58'- a're slidably supported for longitudinal adjustment. Each of the cells 51 and 5B are provided With spherical seats 59 and 60 into which corresponding spherical members -6-I and 62 attached to the rear surfaces of the concave gratings 5I and 52 are v adapted to rest. The said spherical members 6I and 62 are urged into rm sliding contact with the spherical seats 59 and 'll by means of a plurality of adjusting screws, as shown at 63, and 64, whereby the positions of said concave gratings may be universally adjusted within the required angular limits.

-Ther concave. diffraction .gratings` ,5I/ and" 52 may be: of anyY desiredi design suchzasvthat conij ventionally employed,` which, ,for example, usually comprises aglass'blanlr, the concavel figured sur:- face of which has. beenzcoated with athinlayer of aluminum or; the; like suitable. metal` applied by suitable means 'such as by evaporation-and onwhich the'y desired number. of uniformlyspaced parallel lines vhas`- been. ruled by:- meanso fj a diamond.. v y f In theswall 65 of the'f-erwardf end;r portion 23 of thespectograph'housing at. a positionfoppsite 4the.gratings and adjacentrthe slit assemblyis-an elongated, rectangulal1-*shaped lwindow- E', over which a lm-supportingpcamera assembly-is suit.- ablys attached; as .bestshown'at ti'I` .in Figuren-1; The. nlm-holden assembly comprises a, rectangulari frame 68: havingformedmn: the inside a-,pair of' arcuate shoulders yEi-E,y lyingfinwthef sur-f ace of.. an imaginary cylinder lli, hereinafter-described in connection: with 'Figure 4', and a. pressureV fpl ate 11i! or,correspending,l curved shape: is adapted-tancia a stript of l lnr between. the said'. pressure-plate 0Tandithesaidzshould'ers 5:9astshoWnf at ,'IzI'.'l VThe pressure plate .'IiL-` is. adaptedf to,l forced" into clamping; engagement with'. the: frlmf v'I Ir bymeans ofsuitableleafsprings L2 carried' on theeinnerfsi'de of.v a film-.holder back 13 which is', in turn, removably-'attached lto' the, film-holder' assemblyl by means'ofsuitablethumb sicreWsn114--` Although. several types L of: mountingr` .arrangements yof' the f elementsofathe spectograph .may he employed; the type of 'arrangement' herein illus:-

tratedf in. Figures' Iztovl, inclusive; is f arr adaptation ofthat which, infgeneralfollows:.tlieis-clfierneorigiL natedby Rowland for'concavegratings; Asorigi'- vnallyL .conceived by R'owllan-df.the4 elementsof' this y single slit, fthe plurality -oficonc'ave gratings `and the common suracefuporrwhi'ch the" spectra are proj ected, whichsufrface inthe .present-case, yi's-tliek beforemen'ti-on ed' '.sl'ie'etillustrated" at -I-I are all-effectively positionedi toV lie` fin.` an imaginary cylindrical surface, -coaxialfwith and' havin-g va diamet'erequarlto theRowlandl circle, as diagram- Figures 3 and 4. For convenienca/the beforereferred to herein as the Rowland cylinder. Ordinarily, in the conventional arrangement and in the absence of a beam splitter, l.such as that illustrated at 33, the slit 43 would be located, as shown in dotted lines at 43 in Figure 3, on the Rowland circle 1l or lying within the Rowland cylinder `shown in Figure 4. However, when employing a beam splitter vas shown at 33, in order to provide parallel paths of illumination to the separate diffraction gratings, it is necessary .to move the slit 43 inward from the Rowland cylinder a distance d, equal to the lateral deviation i of the beams in the beam splitter, between the diagonal mirrors 44 and vMi, for example, lin order toppreserve the virtual position of the slit within the socalled Rowland cylinder Vl and thus at a total effective distance from `the slit to the gratings, equal to that which it would have in the absence of the beam splitter, and when positioned at 43 on the said Rowland cylinder.

In the Rowland type of mounting, as before stated, the concave diffraction grating has a concave spherical surface, the extended curvature of which is indicated at 16, having a radius of curvature equal to the diameter of the Rowland circle l1, and positioned tangent at l to the Rowland circle 'l1 in .the manner `best shown in Figure 3. The surface upon which the spectrum is projected, for example the curved photographic film, as illustrated at 1I in Figure 3, may be cyl-in-drical in form, and is coaxial with and has a radius of curvature vequal to the radius of fthe said Rowland circle. Thus, with two or more concave gratings arranged as shown at and 52, the slirt 43, and the nlm-holder 61, including the nlm strip 1l are positioned with respect to one `another in the construction illustrated in Figure 1, substantially in accordance with the schematic diagrams Iof Figures 3 and 4. With this arrangement, as will be evident, the film-holder and the gratings will be substantially diametrally opposite one another with the light beam axis lines 19 and I9 practically normal to the surfaces of the gratings 5| and 52 at their cen-ters, for normal spectra. The beam axes 41 and 4l extending from the beam splitter 33 will also intersect the lines 'I9 and 19 at the centers of the concave grating surfaces at their points of -t'angency 18 with the Rowland cylinder Tl, each making a horizontal angle (a) with the respective axes 19 and 19', which angle may have a suitable val-ue dependent upon the dimensions of the instrument, the number of lines in the gratings, the angle of optimum reflectivity of the gratings, and the order of spectra to be obiserved, as hereinafter more fully described.

Referring now again to Figures 1 to 4, inclusive, and particularly, to Figure 4, it has been found that i-f the number of lines per inch of one of the gratings is twice that of the other grating, two spectra of the same source of illumination in different spectral regions may `be simultaneously formed on the same film strip 1I, as illustrated at 80 and 80. For example, a spectrograph 'of the type hereinbefore described was constructed with a grating corresponding to lthat shown at 5| having 12,000 lines per inch and with that correspon-ding to grating 52 having 24,000 lines vper inch, both of the gratings having radius of curvatures of 150 centimeters. The angle (a) was approximately 16 26. With this arrangement, the spectrum produced at 80 from the grating 5I covered the spectral region of from 4,000 to 8,000 A with a dispersion of approximately 14 A per'millimeter, while the adjacent spectrum simultaneouslly produced at 80' from the grating 52 on the same surface, covered the spectral region of from 2,000 :to 4,000 A with a dispersion of approximately 7 A per millimeter. The length of the common film sheet 'il containing the spectra 80 and 80 was approximately 12 inches. Thus, it is seen that by employing .twin gratings, such as that illustrated at 5i and 52, in which the number of lines per unit length of one grating is double that of the other, adjacent spectra may be simultaneously obtained and recorded on a common cylindrical film surface, or, if desired, upon separat-e film surfaces lying adjacent to one another on a common cylindrical surf-ace, the ranges of which are complementary and have dispersions which bear direct ratios one to the other.

While, in the apparatus hereinbefore shown and described, two gratings are shown, any desired number of gratings may be similarly employed to cover various complementary or extended portions of the spectrum with any degree of dispersion as desired.

When a plurality of gratings are thus employed, the number of lines per unit length of grating for each grating may bear any suitable ratio one to another depending upon the relative degree of dispersion required. For example, when, instead of two complementary spectra as illustrated herein, three complementary spectra are required, the

number of lines per unit length of gratings may bear the relationship of 1:2:4, respectively, with the result that three parallel spectra can be formed on a lcommon so-called focal plane surface in a manner similar to that illustrated for two spectra, as illustrated in Figure 4, but with dispersion relationships bearing corresponding ratios. y

For example, three vertically spaced concave diffraction gratings may be employed having 7,000, 14,000, and 28,000 lines per inch, respectively, with the resultant complementary portions of the spectra covering the spectral regions of 7,000 to 14,000; 3,500 to 7,000 and 1,750 to 3,500 and bearing the corresponding relative dispersions of approximately 24 A, 12 A and 6 A, respectively.

Vhe'n a series of gratings are employed as just 'mentioned each succeeding grating in the series may have its number of lines either twice or onehalf of that of the preceding one and thus the spacings S, of the lines for different diffraction `gratings would bear the following relation to the number of individual gratings:

where n, `equals any number from l to the total number of gratings employed, and d, equals a given spacing between lines of one of the gratings.

While, for purposes of illustration, this invention has been described in connection withor as a modification of, the Rowland type of mounting, the invention is also similarly adaptable to several other well-known types of mountings. For example, in Figures 5 and 6, another suitable alternative type of mounting arrangement is shown, to which this invention is readily applicable. In this arrangement, which is an adaptation of the simple Wadsworth type of mounting, the

diffraction gratings are designed to operate from parallel light instead of diverging light as `with the Rowland type and for this purpose, the slit43 is located at the equivalent concurrent focal points of a pair of concave collimating mirrors and 8l. The resulting parallel rays 15 and 15", reflected from the collimating mirrors 80 and 8|, are projected onto the concave figured surfaces of 7 apairof concave' gratings 83 and 841respectively', and the' resulting` spectra projected, as shown at 86. and 02', upon a pair' of curved film strips or a single curved film 85, saidcurved film strip or strips being located: coextensive with the so-called curved focal planes of the gratings.

`In Figures 5 and @,as before stated, anadaptationI of! thev simple Wadsworth mounting has been effectedin accordance with this invention. However, if desired, the mountingcan be further modiied' to position the spectral and the, film-holder, if? any, adjacent the sliti for more convenient operation by a single operator, by employing an additional plane mirror to intercept theY parallel rays at some point on the axis I5-l5 intermediatethe mirrors 80, 8i, and gratings 83, 84. An additionalflight passagel across the spectrograph is' thus, provided which permits the surface 85 to bte'. positioned onv the side of the spectrograph occupied by the slit e3. An example of a suitable construction of this Wadsworth type of twin mountingscomprises concave gratings 83 and 84, 'having V12,000 lines and 24,000 lines per inch, respectively, on e. radius of curvature .of 300 centimeters. The resulting spectra 85 and 8l will coverstigmatically, thel spectral regions of from L000 to 8,000 A and fromv 2,000 to 4,000 A, respec- 1 tively, with dispersions of approximately 13.6 and '618A per millimeter.

In the Rowland adaptation, hereinbefore described, it will be' noted that, that portion .of the spectrum extending into the ultra-violet is derived from that. portion lll of the beam from the beam splitter which is reflected from the surface of the .partial'reector dll. That portionof the spectrum which is confined to the visible region is derived from thebeam lwhich Ipasses through the partial reflectorl d4, which may be made of glass and may' thus serve 'as a filter to absorb the second order. of the' ultra-violet spectrum. In the Wadsworth adaptation, a suitable ultra-violet lter 'glass may be interposed at some convenient point such as in one of the geams i5 or l5 to exclude the'ultra-violet from the visible spectrum.

The twin or plural mounting of gratings, in accordance with this invention as herebefore ldescribed, is not only applicable to those employing concave gratings, but is also similarly applicable to.` ther various well-known mounting arrangements employing pla-ne gratings.

beams; aplurality of concave diffraction gratings, .all having the same focal length and positioned .with a common focal plane, and adapte-d to receive said parallel beams of radiant energy; and means in said focal plane to receive simultaneously spectra from said gratings.

2; A device of the character described, coinprising: a slit-source of a beam of radiant energy having divergent rays; means to receive and to transform said beam into a plurality of like parallel beams; a plurality of concave diffraction gratings, all having the same focal length and positioned parallel with said slitand with a cornmon focal plane, and adapted. tok receive said parages 3g rea allel beams offradiant energy; and means insaid focal plane to receive simultaneously spectra from said gratings.

3. A device of the: character described, comp `ising: a slit-source of a beam of radiant energy having diver-gent rays; means to receive and to transform said beam into a plurality of like parallel beams; a plurality of concave diffraction gratings, allhaving predeterminedv different line spacings, but the same focal lengths and positioned parallel with said slit and with'a common focal plane, and adapted tofreceive said `parallel beams of radiant energy; and means-in said focal plane toreceive simultaneously spectra from'said gratings.

4. A device of the character described, comprising: a slit-source of divergent radiant energy lyingvirtually in' an imaginary cylindrical sur'- face; a plurality ofv concave diffraction gratings spaced from one another'lcngitudinally aloner said cylindrical surface and each having a radius of curvature equal to the diameter of said cylindrical surface and lying tangent to said cylindrical surface in positions adapted simultaneously toreceive said radiant energyjfrom said slitrsource; uniformly spaced lines on said diffraction gratings` with the lines of different gratings having predetermined different spacings and surface means coextensive with saidV cylindrical surface to receive focused, slit spectra fromsaid gratings. i 5. A device of the character described,` comprising: means` for forming, from a single source, a plurality of straight, longitudinally spaced, coaxially alignedslit-sourcesof kradiant energyv lying virtually wholly in an imaginary cylindrical surface; a plurality of concave, diiractiongratings spaced from one-anotherlongitudinally along said cylindrical surface andeach 'having a radius of curvature equal to the diameter of said cylindrical surface and lying. tangent to said cylindrical surface at positions opposite said virtualslitsources and` adapted simultaneously to receive radiant energy therefrom, said diifraction gratings yhaving Idifferent grating line spacings;k and surface means` coextensive with said cylindrical surface to receive focused, slit spectra from said gratings.

6. A deviceY of the character describedgcom.- prising: means for forming, from a single source, a plurality of straight, longitudinally spaced, coaxially aligned, slit-sourcesv of radiant energy lying virtually Wholly in an yimaginary cylindrical surface; a plurality of concave, diifraction gratings spaced from' one another longitudinally along said cylindrical surface and each `having a radius of curvature equal to the ldiameter of said cylindrical surface and lying tangent to the said cy'- lindrical surface at positions opposite said'virtual slit-sources and adapted simultaneously to receive radiant energy therefrom; uniformly spaced lines cylindrical surface and each having a radius of curvature equal to the diameter of said cylindrical 'surface and lying tangent to the said cylindrical surface at positions opposite said slitsource and adapted simultaneously to receive radiant energy therefrom; lines on said diffraction gratings, the lines on one of said gratings having a spacing twice that of the lines on the other of said gratings; and surface means coextensive with said cylindrical surface to receive focused, slit spectra from said gratings.

8. A device of the character described, comprising: a slit-source of a divergent radiant energy beam; means for receiving the divergent radiant energy beam from said slit-source and for forming a plurality of beams therefrom having parallel axes; a plurality of concave, diffraction gratings spaced from one another longitudinally along an imaginary cylindrical surface, each grating having radii of curvature equal to the diameter of said cylindrical surface and lying tangent to the said cylindrical surface at a position adapted to receive one of said parallel beams from said means said diffraction gratings having different grating line spacings, and surface means cceXtensive with said cylindrical surface to receive focused, split spectra from said gratings.

9. A device of the character described, comprising: a slit-source of a divergent radiant energy beam; a beam splitter for receiving the divergent radiant energy from said slit-source and for forming a plurality of beams therefrom having parallel axes; a plurality of concave, diffraction gratings spaced from one another longitudinally along an imaginary cylindrical surface, each grating having radii of curvature equal t0 the diameter of said cylindrical surface and lying tangent to the said cylindrical surface at a position adapted to receive one of said parallel beams from said beam splitter said diffraction gratings having different grating line spacings, and surface means coeXtensive with said cylindrical surface to receive focused, split spectra from said gratings.

10. A device of the character described, in ac- Number cordance with claim 8, in Which the lines of one of said diffraction gratings have double the spacing of the lines of another of said diffraction gratings.

l1. A device of the character described, in accordance With claim 8, in which the lines of one of said diffraction gratings have double the spacing of the lines of another of the diffraction gratings, and the two diffraction gratings are positioned to project corresponding spectra onto the said cylindrical surface in adjacent, longitudinally spaced relationship, such that the Wave length scale of one spectrum is double that of the other.

12. A device of the character described, in accordance with claim 9, in which the lines of one of said diffraction gratings have double the spacing of the lines of another of said diffraction gratings.

13. A device of the character described, in accordance with claim 9, in which the lines of one of said diffraction gratings have double the spacing of the lines of another of the diffraction gratings, and the two diffraction gratings are positioned to project corresponding spectra onto the said cylindrical surface in adjacent, longitudinally spaced relationship, such that the wave length scale of one spectrum is double that of the other.

POLYDORE F. SWINGS.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Name Date Richardson Nov. 12, 1940 FOREIGN PATENTS Country Date Germany Dec. 27, 1939 OTHER REFERENCES Scientific American, March 1944, pages 140 and 141. (Copy in Scientific Library, U. S. Patent Office.)

Number Certificate of Correction Patent No. 2,453,164. November 9, 1948;

POLYDORE F. SWINGS It is hereby certified that errors appear in the printed specification of the above numbered patent requiring correction as follows:

Column 7, line 43, for geams read beams; column 9, line 24, after the Word means insert a comma; lines 27 and 44, for split read slit; line 41, after splitter insert a comma;

and that the said Letters Patent should be read with these corrections therein that the same may conform to the record of the case in the Patent Office.

Signed and sealed this 5th day of April, A. D. 1948.

THOMAS F. MURPHY,

Assistant Commissioner of Patente. 

